Chapter 10
Radicals,
Radical
Functions, and
Rational
Exponents
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Section 10.1
Radical Expressions and
Functions
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Objectives
1. Evaluate square roots.
2. Evaluate square root functions.
3. Find the domain of square root functions.
4. Use models that are square root functions.
5. Simplify expressions of the form a 2 .
6. Evaluate cube root functions.
7. Simplify expressions of the form 3 a 3 .
8. Find even and odd roots.
9. Simplify expressions of the form
n
an .
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Radicals
In this section, we introduce a new category of
expressions and functions that contain roots.
The reverse operation of squaring a number is
finding the square root of the number.
The symbol
that we use to denote the principal
square root is called a radical sign. The number
under the radical sign is called the radicand.
Together we refer to the radical sign and its
radicand as a radical expression.
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Example
Index of the
Radical
n
Radical
Sign
a
Radicand
Radical
Expression
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Definition of the Principal Square Root
If a is a nonnegative real number, the nonnegative
number b such that b2 = a, denoted by b  a , is
the principal square root of a.
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Example 1: Evaluating Square Roots
(1 of 2)
Evaluate:
a. 64
64  8 because 8  64
2
b.  49
 49  7 because 7 2  49
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Example 1: Evaluating Square Roots
(2 of 2)
Evaluate:
c. 16
2
16 4
25
 4  16
 because   
25 5
 5  25
d. 9  16  25
5
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Example 2a: Evaluating Square Root
Functions
For the function, find the indicated function value:
f ( x)  12 x  20; f (3)
f (3)  12(3)  20
 36  20
 16
4
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Example 2b: Evaluating Square Root
Functions
For the function, find the indicated function value:
f ( x)   9  3 x ; g (5)
f (5)   9  3(5)
  9  15
  24
 4.90
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Radical Functions Domain
We have seen that the domain of a function f is
the largest set of real numbers for which the value
of f(x) is a real number. Because only nonnegative
numbers have real square roots, the domain of a
square root function is the set of real numbers for
which the radicand is nonnegative.
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Example 3: Finding the Domain of a
Square Root Function
Find the domain of f ( x)  9 x  27.
The radicand 9x – 27 must be nonnegative.
9 x  27  0
9 x  27
x3
The domain of f is 3,   .
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Example 4: An Application (1 of 2)
Police use the function f ( x)  20 x to estimate
the speed of a car, f(x), in miles per hour, based on
the length, x, in feet, of its skid marks upon sudden
braking on a dry asphalt road. Use the function to
solve the following problem.
A motorist is involved in an accident. A police
officer measures the car’s skid marks to be 45 feet
long. Estimate the speed at which the motorist was
traveling before braking. If the posted speed limit is
35 mph and the motorist tells the officer she was
not speeding, should the officer believe her?
Explain.
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Example 4: An Application (2 of 2)
Substitute 45 for x.
f ( x)  20 x
Use the given function.
f ( x)  20(45)
Substitute 45 for x.
f ( x)  900
Simplify the radicand.
f ( x)  30
Take the square root.
The model indicates that the motorist was traveling
at 30 miles per hour at the time of the sudden
braking. Since the posted speed limit was 35 miles
per hour, the officer should believe that she was
not speeding.
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Simplifying
a2
For any real number a,
a2  a .
In words, the principal square root of a2 is the
absolute value of a.
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Example 5: Simplifying Radical
Expressions
Simplify each expression:
a. (7) 2  7  7
b.
( x  8) 2  x  8
c. 49x10  (7 x 5 ) 2  7 x 5 or 7 x 5
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Definition of the Cube Root of a
Number
The cube root of a real number a is written 3 a .
3
a  b means that b3  a.
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Example 6a: Evaluating Cube Root
Functions
For the function, find the indicated function value:
f ( x)  3 x  6; f (33)
f (33)  3 33  6
 3 27
3
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Example 6b: Evaluating Cube Root
Functions
For the function, find the indicated function value:
g ( x)  3 2 x  2; g ( 5)
g (5)  3 2( 5)  2
 3 8
 2
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Simplifying
3
a3
For any real number a,
3
a 3  a.
In words, the cube root of any expression cubed is
that expression.
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Example 7: Simplifying a Cube Root
Simplify:
3
27x 3  3 (3 x)3
 3x
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Example 8: Finding Even and Odd
Roots
Find the indicated root, or state that the expression
is not a real number.
a.  4 16  2
b.
4
16 is not a real number
c.
5
1  1
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Simplifying
n
an
For any real number a,
1. If n is even, n a n  a .
2. If n is odd,
n
a  a.
n
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Example 9: Simplifying Radical
Expressions
Simplify:
a. ( x  6)  x  6
4
4
5
b. 5 (3 x  2)  3 x  2
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